Comparisons between Multivariate Linear Models
Compute a (generalized) analysis of variance table for one or more multivariate linear models.
# S3 method for mlm anova(object, …, test = c("Pillai", "Wilks", "Hotelling-Lawley", "Roy", "Spherical"), Sigma = diag(nrow = p), T = Thin.row(proj(M) - proj(X)), M = diag(nrow = p), X = ~0, idata = data.frame(index = seq_len(p)), tol = 1e-7)
- an object of class
- further objects of class
- choice of test statistic (see below). Can be abbreviated.
- (only relevant if
test == "Spherical"). Covariance matrix assumed proportional to
- transformation matrix. By default computed from
- formula or matrix describing the outer projection (see below).
- formula or matrix describing the inner projection (see below).
- data frame describing intra-block design.
- tolerance to be used in deciding if the residuals are
anova.mlm method uses either a multivariate test statistic for
the summary table, or a test based on sphericity assumptions (i.e.
that the covariance is proportional to a given matrix). For the multivariate test, Wilks' statistic is most popular in the
literature, but the default Pillai--Bartlett statistic is
recommended by Hand and Taylor (1987). See
summary.manova for further details. For the
"Spherical" test, proportionality is usually with the
identity matrix but a different matrix can be specified using
Corrections for asphericity known as the Greenhouse--Geisser,
respectively Huynh--Feldt, epsilons are given and adjusted \(F\) tests are
performed. It is common to transform the observations prior to testing. This
transformation to intra-block differences, but more complicated
within-block designs can be encountered,
making more elaborate transformations necessary. A
T can be given directly or specified as
the difference between two projections onto the spaces spanned by
X, which in turn can be given as matrices or as
model formulas with respect to
idata (the tests will be
invariant to parametrization of the quotient space
M/X). As with
anova.lm, all test statistics use the SSD matrix from
the largest model considered as the (generalized) denominator. Contrary to other
anova methods, the intercept is not excluded
from the display in the single-model case. When contrast
transformations are involved, it often makes good sense to test for a
An object of class
"anova" inheriting from class
The Huynh--Feldt epsilon differs from that calculated by SAS (as of v. 8.2) except when the DF is equal to the number of observations minus one. This is believed to be a bug in SAS, not in R.
Hand, D. J. and Taylor, C. C. (1987) Multivariate Analysis of Variance and Repeated Measures. Chapman and Hall.
require(graphics) utils::example(SSD) # Brings in the mlmfit and reacttime objects mlmfit0 <- update(mlmfit, ~0) ### Traditional tests of intrasubj. contrasts ## Using MANOVA techniques on contrasts: anova(mlmfit, mlmfit0, X = ~1) ## Assuming sphericity anova(mlmfit, mlmfit0, X = ~1, test = "Spherical") ### tests using intra-subject 3x2 design idata <- data.frame(deg = gl(3, 1, 6, labels = c(0, 4, 8)), noise = gl(2, 3, 6, labels = c("A", "P"))) anova(mlmfit, mlmfit0, X = ~ deg + noise, idata = idata, test = "Spherical") anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ noise, idata = idata, test = "Spherical" ) anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ deg, idata = idata, test = "Spherical" ) f <- factor(rep(1:2, 5)) # bogus, just for illustration mlmfit2 <- update(mlmfit, ~f) anova(mlmfit2, mlmfit, mlmfit0, X = ~1, test = "Spherical") anova(mlmfit2, X = ~1, test = "Spherical") # one-model form, eqiv. to previous ### There seems to be a strong interaction in these data plot(colMeans(reacttime))